Tuesday, February 05, 2013

Steve Hsu: Alice is Schrödinger's cat who may have fallen to a black hole

...or not...

The black hole firewall saga continues. The original paper by AMPS has collected 35 citations according to SPIRES. Most of the recent 10 are papers that are primarily about issues that different than black hole firewalls.

However, Stephen Hsu who is also a blogger (article about the topic; don't confuse him with Steven Chu although there may be some similarities here, too) just posted a new, 3-page preprint that attacks the essential error done by AMPS:

I believe his basic line of arguments is equivalent to what I've been saying and it's also compatible with the Raju-Papadodimas paper and Nomura-Varela-Weinberg papers. What's the basic fact that Hsu realizes and AMPS overlook?

The most important fact of this sort is that for a fixed pure initial state of the star, Alice (the infalling observer) always has some probability amplitudes (and some overall probability) that she falls into a black hole and some probability amplitudes (and some overall probability) that she doesn't. There are many possible microstates (possible evolution) in the first group and many possible microstates (possible evolution) in the second group and the overall state vector is the sum of all the pieces from both groups. I discussed the presence of the "Yes" as well as "No" branches here (click).

I found this cartoon by a rudimentary Alice-Bob search on Google Images but the content of the cartoon is actually exactly what I need. Alice correctly tells Bob who wants to marry her that in a quantum world, both possibilities – she will marry him and she won't, she will escape him to a black hole interior or she won't – have nonzero probability amplitudes. As the article below discusses many times, Bob much like Polchinski et al. seem to misunderstand this basic point about quantum mechanics. Well, this misunderstanding may be shared by most men as well because they often interpret the quantum mechanical women's "Maybe" as "Yes". :-)

Only the overall wave function must evolve unitarily from the initial state, only the total wave function (including pieces with Alice inside and pieces with Alice never inside any black hole) are subject to the constraint that the information should be preserved. This accuracy is very important because, as Papadodimas and Raju emphasized, even exponentially tiny corrections to the operators (I mean the matrix entries are exponentially tiny) may turn a pure state into a mixed one and vice versa.

Hsu says that Alice observes no firewalls but it's important to discuss "which Alice" we mean when we say that she sees no firewall; and "which Alice[s]" we need to preserve the information. The answers to both questions differ.

The answer to the first question is that a particular Alice with a particular history, one in which she's sure that she's falling into a black hole (a single term in the overall wave function), will experience no firewalls. If we imagined that the whole quantum evolution (evolved state) contains this Alice (term), we could argue – following AMPS – that the information isn't conserved.

However, the unitary evolution of the initial state contains, as we have emphasized, also other "Alices", including "Alices" who avoid the black hole interior at all times. The word "contain" in the previous sentence is meant to represent the mathematical inclusion of a term in a sum; the correct "physical" interpretation says that the laws of quantum mechanics imply that it's possible for Alice to evolve differently and, perhaps, avoid the black hole throughout her life. The total wave function is a state similar to Schrödinger's cat, one composed of macroscopically distinct states that quickly decohere from each other. This state evolves unitarily from the initial state and will evolve unitarily to the final state of the Hawking radiation that remembers the initial state.

There's no need for the information to be preserved on the "branch" of one particular Alice. One particular Alice who fell into a black hole just made some measurements of certain observables \(A_i\) that identify her "branch" of the wave function. One of these observables \(A_0\) is the qubit that determines whether she fell into a black hole or not.

These observables \(A_i\), including \(A_0\), refuse to commute with the observables \(B_k\) that observers outside the black hole (spatially separated from the particular Alice who is inside) may perform. There's nothing wrong about this fact; the fact that these sets of operators \(\{A_i\},\{B_k\}\) don't commute with each other is the essence of black hole complementarity. As long as the probabilities of the measured values of \(A_i\) are nonzero in the eigenstates corresponding to the measured values of \(B_k\), there is no contradiction.

Polchinski et al. made the mistake of trying to restrict the discussion of all the observations (including those outside) to the subspace of eigenvalues \(A_i\) that a particular infalling Alice has measured. But that's just wrong: most of the states identified by other observations (e.g. those done outside the hole, for example by Bob) fail to belong to this low-dimensional space because the relevant states onto which Bob projects are eigenstates of operators such as \(B_k\) that don't commute with \(A_i\), so they can't possibly belong to a particular subspace of shared \(A_i\) eigenstates.

Steve Hsu also discusses one interesting point showing that the fact that the black hole evolves into a superposition of macroscopically distinct states isn't just a formality that affects Alice in the same sense as it usually affects Schrödinger's cat. Instead, the black hole itself is evolving into a superposition of macroscopically distinct states, especially when it comes to the location of the black hole.

When the black hole is sending the Hawking quanta to random directions, their total momentum more or less averages out. But it doesn't average out exactly. The momentum of a Hawking particle goes like the temperature \(T\sim 1/R\) and each particle reduces the entropy of the remaining black hole by something like \(\Delta S\sim -1\). After the Page time (half of the initial entropy has been evaporated away), the black hole has sent something like \(S/2\) Hawking quanta. The momentum of each was \(1/R\) with a random sign/direction. If we add them, we may see by the maths of random walks that the total momentum deposited to the recoiling black hole shooting the Hawking quanta is of order \(\pm\sqrt{S/2}/R\sim\pm 1/\sqrt{G}\) (the last expression only holds for \(d=4\) while the previous expressions held for a general \(d\)) so the average velocity is of order \(P/M=\pm 1/M\sqrt{G}\). Yes, that's smaller than \(c=1\) because the mass of the black hole is larger than the Planck mass \(1/\sqrt{G}\).

When you finally multiply this velocity by the Page time \(t\sim G^2 M^3\), you get the estimate for the total distance that the black hole has traveled after the long Page time due to the recoils as it was shooting the Hawking bullets:\[

\] Now, all the formulae are only OK in \(d=4\). In the Planck units, the change of the location of the black hole goes like \(M^2\) which is still a very high number i.e. long distance! It is much longer, \(M \sqrt{G}\) times, than the black hole radius. (I wrote the derivation directly into the blog and haven't seen it previously; thank God, the result agreed with Hsu's underived claim.) So the Hawking quanta are pretty random but they easily combine to an uncertainty of the black hole location that may become macroscopic – in fact, much longer than the black hole radius – after a sufficiently long time such as the Page time.

(If you want to know, after the fraction \(f\) of the black hole mass evaporates away, \(\Delta x\) is generalized to \(f^{3/2}G^{3/2}M^2\); the simplified expression above was for the Page fraction \(f=1/\sqrt{2}\). It's not hard to derive the \(f^{3/2}\) dependence, I think: the velocity goes like \(f^{1/2}\) for the same reason why the random walk \(\Delta x\) goes like \(t^{1/2}\); the power \(f^{3/2}\) is just the indefinite integral of that which adds \(1\) to the exponent. Also, note that the "branches" with different locations of the black hole inevitably decohere from each other if we decide to trace over the degrees of freedom in the outgoing Hawking radiation.)

It's very important not to overlook the other branches of the evolution, including the inevitably nonzero branches in which Alice never falls into a black hole, if we want to verify that the information is conserved. AMPS failed in this test of accuracy. They were imagining that the full exact quantum evolution contains "one particular Alice with one particular life story" which is one of the typical errors in which people incorrectly use a classical reasoning in the quantum realm or, if I locate the blunder even more precisely, the error in which people "erase" the other terms in the wave function ("collapse" the wave function) prematurely because they just feel uncomfortable about the superpositions and want to return to the classical world (where all properties are objectively and uniquely determined) as soon as possible. But in this case (much like others), the "collapse" is premature, indeed, because the preservation of the information may depend on the interference of the many branches in which Alice fell into a black hole and the branches in which she hasn't.

(In fact, AMPS – and others – are not only making the error of assuming that Alice's being inside a black hole is a classical fact. They often want to determine Alice's position relatively to the horizon with quite some amazing precision. This contradicts the inevitably inaccuracy resulting from the random-walk character of the Hawking evaporation. The stunning similarity of this mistake to Einstein's mistake during his debates with Bohr – those about Einstein's box – suggest that physicists have learned almost nothing from these debates.)

So yes, Hsu's paper is another one that helps to settle my original suspicion that the error done by Polchinski et al. ultimately boils down to their misunderstanding of the foundations of quantum mechanics, something that they (much like others) try to "reshape" far too classically.

And that's the memo.

P.S.: I can't resist to mention that in 2005, when Stephen Hawking admitted that the information wasn't lost after all, he also offered his own proof that it doesn't. When properly reorganized, most of the histories that contribute to the final Hawking radiation are histories in which the black hole is completely avoided, he argued. This 2005 claim due to Hawking (which wasn't uncritically accepted as "yet another full proof", just to be sure) is not quite the same thing as the claim in Hsu's paper – that it's important that Alice herself may avoid the black hole interior – but it also has a similar spirit because it emphasizes the importance of histories and observers that never see a/the black hole interior. One may only derive the paradoxical "information is lost" conclusion if he assumes that the chances are 100% that the black hole is formed and an observer sees the interior but the probability that this ain't so, while it may be small, is always important to fix the qualitative conclusion and to see that the information is preserved.

Oh, if only we still had Carl Sagan with us, we could use his uncanny ability to explain big numbers to regular people like me.

He would start out by explaining how, thanks to Benny and the Inkjets, by 2020 there will be more dollars in circulation than there are grains of sand on all the beaches of the world. And by 2030 there will be more dollars than there are atoms in the entire universe. There will be so many dollars in circulation that their gravitational field, while small individually, is enough to cause a financial black hole from which no asset can escape- not even the barest few photons of light from a brand-spanking-new iPhone 5!

However, the term "histories" for the different possible stories or branches confuses me a bit, are these histories in the sense of "consistent histories"?The whole argument reminds me to the fact that one should consider all paths when using path integrals to calculate transitions between two states, instead of only taking those with Alice falling into the black hole into account for example (?) ...

The calcution of the position uncertainty of a black hole is cool too and new to me :-)

It is nice to see you have finally accepted the many worlds interpretation of quantum mechanics. Everyone knows that you cannot ask "which Alice" without implying there exists a state space in which all Alices live, and that the name of this space is the MWI. Now that you have come to your senses maybe you can convert Polchinski.

Jesus! To anyone with even a rudimentary knowledge of English grammar it sounds just awful! You cannot ever mix tenses, not ever! Even a kindergartener would be immediately corrected by the teacher, who would immediately conclude that the child’s parents have zero education. This is not a subtle thing!

It is hard, really hard, to think entirely probabilistically. Even the smartest of us is tempted by the devil “reality” and many of the best and brightest succumb to that temptation. This lure has led thousands of brilliant minds into the many worlds’ delusion or to the idea that something is wrong with QM. That is another delusion, of course.

Lubos understands this at a very deep level and those TRF readers who don’t need to read TRF carefully, very carefully.

I haven't accepted any "many worlds interpretation" which is ill-defined nonsense. There exist (in the mathematical sense, in the Hilbert space) states with different Alices, states where Alice has different properties. But that doesn't mean they exist in the real world, simultaneously. Instead, the right interpretation is that quantum mechanics says that Alice may have different properties and location with different probabilities. I wrote this point very explicitly in this blog entry, too. But certain people prefer their idiotic preconceptions and they never abandon their totally foolish belief that the laws of physics must suddenly change at every moment so that their misconceptions become true, right? They won't.

Lol. My kids keep correcting my English pronunciation and I don't mind. The other day, they burst laughing because I was saying "focus" with a too strong intonation on the first syllable which sounded like "fo'ckus". It should sound like "fowc's" with a long first syllable. Sight. Some words I have to take a pause before saying them :-).

Dear Dilaton, "histories" always means the same rough thing, informally - collections/lists of events/observations in a region of spacetime, one possible way how a snapshot of a region of spacetime may look.

However, these histories aren't consistent histories which are supposed to be classically interpretable, typically decohered from each other, histories. Instead, here we are talking about histories in the most accurate possible quantum sense - they're really Feynman histories such as those from "Feynman's sum-over-histories approach to quantum mechanics".

Dear Gene, face the harsh reality. When I write "have fell", it's not because I would be mixing tenses in some advanced way. The error is simply because I don't remember reliably enough what the past participle exactly is. For many verbs, it's just the same word as the past simple form.

We would be memorizing things like fall/fell/fallen, and so on. But there's just so much of this useless stuff.

Dilaton - Perhaps not as good as the legendary Italian immigrant who got nervous when a cop stopped him. The cop asked him to spell his name. The Italian, between nervousness and a shaky grasp on the English alphabet, said, "EM EYE ESS SEE MY ESS SEE MY OH."

I certainly did not mean to criticize your English skills, Lubos, which are truly remarkable. I have read thousands of your blogs and cannot recall any ambiguity in your use of English expressions whatsoever. I never have to read your prose twice to get your meaning. That, my friend, is amazing!

When I studied English more than sixty years ago there was great emphasis on getting the tenses right as well as all of the irregular forms in the English language, which are ubiquitous. I ought to have simply stated that ”have fell” sounds just awful to my ear. I had tough teachers and I am uncomfortable with the relaxation of standards these days.

Dear Gene, my English skills may be classified as mediocre and this typo is a sign of that - even though I actively knew that it was a wrong form, of course. Such things are just not really hardwired in my brain.

Some people talk about producing language subconsciously. I think I never do that. It seems to me that the same part of brain must be active when I try to write down "have fallen" as the piece of brain that does the Feynman path integrals. At least I can't do both things at the same moment, via multitasking! :-)

Maybe I belong to an extinct species, Smoking Frog, but I if “have fell” is not immediately corrected by parent or teacher there has occurred, in my view, an unacceptable relaxation of standards. “Have fell” is just terrible!

When Feynman learned enough Portuguese to lecture in Brazil he did not master the language and, no doubt, made hundreds of mistakes. I would wager that his audience understood his meaning perfectly, however, and was polite enough to ignore those errors. Clarity of thought is one thing and precise grammatical form quite another. I guess I should be more polite!

If the first syllable in “fowc’s” rhymes with “cow” it is still incorrect. It should rhyme with “low”. English is a bitch, isn’t it? The classical example is the vastly differing pronunciation of the four English words, “cough”, dough”,“tough” and “bough”. Yep, a bitch.

Gene, French is more simple in a way. All syllables are pronounced entirely and equally with no emphasizing needed on some syllables. My worse English words to understand are : I "can" and I "can't". My French ear doesn't seem to distinguish the difference unless it is said in a "proper British" way (kind of the posh way... kan'tttth).

The “t” in “can’t” is not silent. Anyone who fails to voice it is being sloppy and I am not big on sloppy, as you know. Lazy people rely on context to distinguish “can” from ”can’t”. Sometimes they use a glottal stop (brief interruption of the breath) instead of voicing the “t” but that’s still laziness. Screw ‘em.

SmokingFrog, it's quite the opposite. We think you are barbarians :~) who don't bother with consonants :-). For example British would say "toma'oes, pota'oes, grea'"dropping the "t". Is this why you drink so much tea ;^) ... (my early morning joke). Is T a posh consonant on your island I wonder ?(Here in Ireland they don't use the T. They have replaced it by CH or SH).

---a Feynman-Gell-Mann story:Feynman once encountered Gell-Mann in the hall outside their offices at Caltech and asked him where he had been on a recent trip; "Moon-TRAY-ALGH!" Gell-Mann responded in a French accent so thick that he sounded as if he were strangling. Feynman -- who, like Gell-Mann, was born in New York City -- had no idea what he was talking about. "Don't you think," he asked Gell-Mann, when at length he had ascertained that Gell-Mann was saying "Montreal," "that the purpose of language is communication?"

Shannon - I've changed my mind a little. I now think you're at least partly right about spoken English and the consonants. I had forgotten that I've noticed a need to pronounce some consonants more "emphatically" in Spanish. However, I think the deficiency is greater in the not-so-posh classes in England and possibly Ireland than it is here, and of course that's where you are. Also, many Mexicans and Mexican-Americans do a poor job of pronouncing the English consonants, which contradicts what I just said about Spanish. I don't know how to resolve that paradox.

OTOH, a friend of mine who has actually studied French readily agreed with my initial idea. He's no expert, but this has to count for something. But I suspect he over-agreed.

Yes, I'm aware of the CH, e.g., Chuesday.

Here's an oddity: The Mexican nickname for "Jesus" is "Chuy" (CHEWY). Makes me think of the Communion host. :-)

As a scientist do you not find that one's ability can be transferred to different sectors and subjects, so as to believe that this perspective as a scientist may shine a light where no light shone before?

The light, is the understanding that cross pollination between subjects allows perspective about such a thing as the economy to suggest that one's science when properly portrayed may allow new insights that had not been previously seen before.

Schrodinger may be thrown out as a tidbit so as to suggest that the future may share some view respective of a "Quantum biology" so as to trace its history back to him and the development forward?

Stephen Hsu while dealing with the subject of the economy may have tired and wanted to return to the subject of his science of what he may holds most dear?

Such an idea as to "save the economy," while seeing the tendency in the political realm, as to its faults and distinctions signaled the course and direction of the economy needing an adjustment. The economy, as needing a new course and direction so as to assign it the importance of addressing what had caused the economy to flail and slumber under such a regime?:)